Understanding Compound Angles

Compound angles involve the addition or subtraction of two angles. In trigonometry, the compound angle formulas are essential for simplifying the expressions that include the sum or difference of two angles. These formulas are particularly useful in solving trigonometric equations, proving identities, and performing calculations in geometry and physics.

Compound Angle Formulas

The basic compound angle formulas for sine, cosine, and tangent are as follows:

Sine of a Compound Angle

The sine of the sum of two angles (α + β) and the sine of the difference of two angles (α - β) can be expressed as:

$$ \sin(α + β) = \sin α \cos β + \cos α \sin β $$

$$ \sin(α - β) = \sin α \cos β - \cos α \sin β $$

Cosine of a Compound Angle

The cosine of the sum of two angles (α + β) and the cosine of the difference of two angles (α - β) can be expressed as:

$$ \cos(α + β) = \cos α \cos β - \sin α \sin β $$

$$ \cos(α - β) = \cos α \cos β + \sin α \sin β $$

Tangent of a Compound Angle

The tangent of the sum of two angles (α + β) and the tangent of the difference of two angles (α - β) can be expressed as:

$$ \tan(α + β) = \frac{\tan α + \tan β}{1 - \tan α \tan β} $$

$$ \tan(α - β) = \frac{\tan α - \tan β}{1 + \tan α \tan β} $$

Differences and Important Points

Here is a table summarizing the differences and important points of the compound angle formulas:

Function	Sum of Angles (α + β)	Difference of Angles (α - β)
Sine	$\sin α \cos β + \cos α \sin β$	$\sin α \cos β - \cos α \sin β$
Cosine	$\cos α \cos β - \sin α \sin β$	$\cos α \cos β + \sin α \sin β$
Tangent	$\frac{\tan α + \tan β}{1 - \tan α \tan β}$	$\frac{\tan α - \tan β}{1 + \tan α \tan β}$

• Symmetry: Notice the symmetry in the formulas. For sine, the sign changes between the sum and difference formulas. For cosine, the sign in the middle changes, while for tangent, the sign in the denominator changes.
• Reciprocal Identities: These formulas can be extended to other trigonometric functions like cotangent, secant, and cosecant using reciprocal identities.
• Special Cases: When α = β, these formulas can be used to derive double-angle formulas, which are useful in their own right.

Examples to Explain Important Points

Example 1: Using Sine Compound Angle Formula

Calculate $\sin(75°)$ using compound angle formulas.

Solution:

We can express 75° as the sum of two angles whose sine and cosine values we know from the standard angles (30° and 45°).

$$ \sin(75°) = \sin(30° + 45°) $$

Using the compound angle formula for sine:

$$ \sin(75°) = \sin(30°)\cos(45°) + \cos(30°)\sin(45°) $$

$$ \sin(75°) = \left(\frac{1}{2}\right)\left(\frac{\sqrt{2}}{2}\right) + \left(\frac{\sqrt{3}}{2}\right)\left(\frac{\sqrt{2}}{2}\right) $$

$$ \sin(75°) = \frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4} $$

$$ \sin(75°) = \frac{\sqrt{2} + \sqrt{6}}{4} $$

Example 2: Using Cosine Compound Angle Formula

Prove that $\cos(90° - α) = \sin α$.

Solution:

Using the compound angle formula for cosine:

$$ \cos(90° - α) = \cos 90° \cos α + \sin 90° \sin α $$

Since $\cos 90° = 0$ and $\sin 90° = 1$, the equation simplifies to:

$$ \cos(90° - α) = 0 \cdot \cos α + 1 \cdot \sin α $$

$$ \cos(90° - α) = \sin α $$

This proves the identity.

Example 3: Using Tangent Compound Angle Formula

Find the value of $\tan(15°)$ using the compound angle formula for tangent.

Solution:

We can express 15° as the difference of two angles whose tangent values we know (45° and 30°).

$$ \tan(15°) = \tan(45° - 30°) $$

Using the compound angle formula for tangent:

$$ \tan(15°) = \frac{\tan 45° - \tan 30°}{1 + \tan 45° \tan 30°} $$

$$ \tan(15°) = \frac{1 - \frac{1}{\sqrt{3}}}{1 + 1 \cdot \frac{1}{\sqrt{3}}} $$

$$ \tan(15°) = \frac{\sqrt{3} - 1}{\sqrt{3} + 1} $$

By rationalizing the denominator:

$$ \tan(15°) = \frac{(\sqrt{3} - 1)(\sqrt{3} - 1)}{(\sqrt{3} + 1)(\sqrt{3} - 1)} $$

$$ \tan(15°) = \frac{3 - 2\sqrt{3} + 1}{3 - 1} $$

$$ \tan(15°) = \frac{4 - 2\sqrt{3}}{2} $$

$$ \tan(15°) = 2 - \sqrt{3} $$

These examples illustrate how compound angle formulas can be applied to calculate trigonometric functions of non-standard angles and to prove trigonometric identities. Understanding and memorizing these formulas is crucial for success in trigonometry and related fields.